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Effective interface conditions for a model of tumour
invasion through a membrane
Giorgia Ciavolella, Noemi David, Alexandre Poulain
To cite this version:
Giorgia Ciavolella, Noemi David, Alexandre Poulain. Effective interface conditions for a model of tumour invasion through a membrane. 2021. �hal-03231456�
Effective interface conditions for a model of tumour invasion
through a membrane
Giorgia Ciavolella123 Noemi David14 Alexandre Poulain1 May 6, 2021
Abstract
Motivated by biological applications on tumour invasion through thin membranes, we study a porous-medium type equation where the density of the cell population evolves under Darcy’s law, assuming continuity of both the density and flux velocity on the thin membrane which separates two domains. The drastically different scales and mobility rates between the membrane and the adjacent tissues lead to consider the limit as the thickness of the membrane approaches zero. We are interested in recovering the effective interface problem and the transmission conditions on the limiting zero-thickness surface, formally derived by Chaplain et al. (2019), which are compatible with nonlinear generalized Kedem-Katchalsky ones. Our analysis relies on a priori estimates and compactness arguments as well as on the construction of a suitable extension operator which allows to deal with the degeneracy of the mobility rate in the membrane, as its thickness tends to zero.
2010 Mathematics Subject Classification. 35B45; 35K57; 35K65; 35Q92; 76N10; 76S05;
Keywords and phrases. Membrane boundary conditions; Effective interface; Porous medium equation; Nonlinear reaction-diffusion equations; Tumour growth models
1
Introduction
We consider a model of cell movement through a membrane where the density population u = u(t, x) is driven by porous medium dynamics. We assume the domain to be an open and bounded set Ω ⊂ R3 which is divided into three open subdomains, Ωi,ε for i = 1, 2, 3, where
ε > 0 is the thickness of the intermediate membrane, Ω2,ε, see Figure1. In the three domains,
the cells are moving with different constant mobilities, µi,ε, for i = 1, 2, 3, and they are allowed to cross the adjacent boundaries of these domains which are Γ1,2,ε (between Ω1,ε and Ω2,ε) and
Γ2,3,ε (between Ω2,ε and Ω3,ε). The system reads as
1
Sorbonne Université, Inria, Université de Paris, Laboratoire Jacques-Louis Lions, UMR7598, 4 place Jussieu, 75005 Paris, France. Emails: giorgia.ciavolella@sorbonne-universite.fr, noemi.david@sorbonne-universite.fr, alexandre.poulain@sorbonne-universite.fr
2Dipartimento di Matematica, Università di Roma "Tor Vergata", Italy 3
Corresponding author
∂tui,ε− µi,ε∇ · (ui,ε∇pi,ε) = ui,εG(pi,ε) in (0, T ) × Ωi,ε, i = 1, 2, 3,
µi,εui,ε∇pi,ε· ni,i+1= µi+1,εui+1,ε∇pi+1,ε· ni,i+1 on (0, T ) × Γi,i+1,ε, i = 1, 2,
ui,ε= ui+1,ε on (0, T ) × Γi,i+1,ε, i = 1, 2,
ui,ε= 0 on (0, T ) × ∂Ω.
(1)
We denote by pi,ε the density-dependent pressure, which is given by the following power law pi,ε= uγi,ε, with γ > 1.
In this paper, we are interested in studying the convergence of System (1) as ε → 0. When the thickness of the thin layer decreases to zero, the membrane collapses to a limiting interface, ˜Γ1,3,
which separates two domains denoted by ˜Ω1 and ˜Ω3, see Figure 1. We derive in a rigorous way
the effective problem (2), and in particular, the transmission conditions on the limit density, ˜u, across the effective interface. Assuming that the diffusion coefficients satisfy µi,ε> 0 for i = 1, 3 and
lim
ε→0µ1,ε= ˜µ1 ∈ (0, +∞), ε→0lim
µ2,ε
ε = ˜µ1,3∈ (0, +∞), ε→0limµ3,ε= ˜µ3∈ (0, +∞),
we prove that Problem (1) converges to the following system ∂tu˜i− ˜µi∇ · (˜ui∇˜pi) = ˜uiG(˜pi) in (0, T ) × ˜Ωi, i = 1, 3, ˜ µ1,3JΠK = ˜µ1u˜1∇˜p1· ˜n1,3 = ˜µ3u˜3∇˜p3· ˜n1,3 on (0, T ) × ˜Γ1,3, ˜ u = 0 on (0, T ) × ∂Ω, (2)
where Π satisfies Π0(u) = up0(u), namely
Π(u) := γ
γ + 1u
γ+1.
We use the symbol J(·)K to denote the jump across the interfaceΓ˜1,3, i.e.
JΠK := γ γ + 1(˜u γ+1) 3− γ γ + 1(˜u γ+1) 1, (3)
where the subscript indicates that (·) is evaluated as the limit to a point of the interface coming from the subdomain ˜Ω1, ˜Ω3, respectively.
Figure 1: We represent here the bounded cylindrical domain Ω of length L. On the left, we can see the subdomains Ωi,ε with related outward normals. The membrane Ω2,ε of thickness ε > 0
is delimited by Γi,i+1,ε = {x3 = ±ε/2} ∩ Ω which are symmetric with respect to the effective
interface, ˜Γ1,3 = {x3 = 0} ∩ Ω. On the right, we represent the limit domain as ε → 0. The
effective interface, ˜Γ1,3, separates the two limit domains, ˜Ω1, ˜Ω3.
Motivations and previous works. Nowadays, a huge literature can be found on the mathe-matical modeling of tumour growth [6,13,16,24,26,28,32], on a domain Ω ⊆ Rd(with d = 2, 3
for in vitro experiments, d = 3 for in vivo tumours). Studying tumour’s evolution, a crucial and challenging scenario is represented by cancer cells invasion through thin membranes. In par-ticular, one of the most difficult barriers for the cells to cross is the basement membrane. This kind of membrane separates the epithelial tissue from the connective one, providing a barrier that isolates malignant cells from the surrounding environment. At the early stage, cancer cells proliferate locally in the epithelial tissue originating a carcinoma in situ. Unfortunately, cancer cells could mutate and acquire the ability to migrate by producing matrix metalloproteinases (MMPs), specific enzymes which degrade the basement membrane. A specific study can be done on the relation between MMP and their inhibitors as in Bresch et al. [31]. Instead, we are interested in modeling cancer transition from in situ stage to the invasive phase.
Since in biological systems the membrane is often much smaller than the size of the other com-ponents, it is then convenient and reasonable to approximate the membrane as a zero-thickness one, as done in [10,14], differently from [31]. In particular, it is possible to mathematically de-scribe cancer invasion through a zero thickness interface considering a limiting problem defined on two domains. The system is then closed by transmission conditions on the effective interface which generalise the classical Kedem-Katchalsky conditions. The latter were first formulated in [18] and are used to describe different diffusive phenomena, such as, for instance, the transport of molecules through the cell/nucleus membrane [9, 12, 34], solutes absorption processes through the arterial wall [30], the transfer of chemicals through thin biological membranes [8], or the transfer of ions through the interface between two different materials [2].
For these reasons, studying the convergence as the thickness of the membrane tends to zero represents a relevant and interesting problem both from a biological and mathematical point of view. In the literature, this limit has been studied in different fields of applications other than tumour invasion, such as, for instance, thermal, electric or magnetic conductivity, [21,33], or transport of drugs and ions through an heterogeneous layer, [25]. Physical, cellular and ecological applications characterised the bulk-surface model and the dynamical boundary value problem, derived in [22] in the context of boundary adsorption-desorption of diffusive substances between a bulk (body) and a surface. Another class of limiting systems is offered by [20], in the case in which the diffusion in the thin membrane is not as small as its thickness. Again,
this has a very large application field, from thermal barrier coatings (TBCs) for turbine engine blades to the spreading of animal species, from commercial pathways accelerating epidemics to cell membrane.
As it is now well-established, see for instance [7], living tissues behave like compressible fluids. Therefore, in the last decades, mathematical models have been more and more focusing on the fluid mechanical aspects of tissue and tumour development, see for instance [3,6,7,10,16,26]. Tissue cells move through a porous embedding, such as the extra-cellular matrix (ECM). This nonlinear and degenerate diffusion process is well captured by filtration-type equations like the following, rather than the classical heat equation,
∂tu + ∇ · (uv) = F (u), for t > 0, x ∈ Ω. (4)
Here F (u) represents a generic density-dependent reaction term and the model is closed with the velocity field equation
v := −µ∇p, (5)
and a density-dependent law of state for the pressure p := f (u). The function µ = µ(t, x) ≥ 0 represents the cell mobility coefficient and the velocity field equation corresponds to the Darcy law of fluid mechanics. This relation between the velocity of the cells and the pressure gradient reflects the tendency of the cells to move away from regions of high compression.
Our model is based on the one by Chaplain et al. [10], where the authors formally recover the effective interface problem, analogous to System (2), as the limit of a transmission problem, (or thin layer problem) cf. System (1), when the thickness of the membrane converges to zero. They also validate through numerical simulations the equivalence between the two models. When shrinking the membrane Ω2,ε to an infinitesimal region, ˜Γ1,3, (i.e. when passing to the limit
ε → 0, where ε is proportional to the thickness of the membrane), it is important to guarantee that the effect of the thin membrane on the cell invasion remains preserved. To this end, it is essential to make the following assumption on the mobility coefficient of the subdomain Ω2,ε,
µ2,ε−ε→0−−→ 0 such that
µ2,ε
ε
ε→0
−−−→ ˜µ1,3.
This condition implies that, when shrinking the pores of the membrane, the local permeability of the layer decreases to zero proportionally with respect to the local shrinkage. The function ˜µ1,3
represents the effective permeability coefficient of the limiting interface ˜Γ1,3, i.e. the permeability
of the zero-thickness membrane. We refer the reader to [10, Remark 2.4] for the derivation of the analogous assumption in the case of a fluid flowing through a porous medium. In [10], the authors derive the effective transmission conditions on the limiting interface, ˜Γ1,3, which relates
the jump of the quantity Π := Π(u), defined by Π0(u) = uf0(u) and the normal flux across the interface, namely
˜
µ1,3JΠK = ˜µiu˜i∇f (˜ui) · ˜n1,3= ˜µi∇Π(˜ui) · ˜n1,3, for i = 1, 3 on ˜Γ1,3.
1
These conditions turns out to be the well-known Kedem-Katchalsky interface conditions when f (u) := ln(u), for which Π(u) = u + C, C ∈ R, i.e. the linear diffusion case.
In this paper, we provide a rigorous proof to the derivation of these limiting transmission conditions, for a particular choice of the pressure law. To the best of our knowledge, this question has not been addressed before in the literature for a non-linear and degenerate model such as System (1). Although our system falls into the class of models formulated by Chaplain et al., we consider a less general case, making some choices on the quantities of interest. First of all,
1
for the sake of simplicity, we assume the mobility coefficients µi,εto be positive constants, hence they do not depend on time and space as in [10]. We take a reaction term of the form uG(p), where G is a pressure-penalized growth rate. Moreover, we take a power-law as pressure law of state, i.e. p = uγ, with γ ≥ 1. Hence, our model turns out to be in fact a porous medium type model, since Equations (4,5) read as follows
∂tu −
γ γ + 1∆u
γ+1 = uG(p), for t > 0, x ∈ Ω.
The nonlinearity and the degeneracy of the porous medium equation (PME) bring several ad-ditional difficulties to its analysis compared to its linear and non-degenerate counterpart. In particular, the main challenge is represented by the emergence of a free boundary, which sep-arates the region where u > 0 from the region of vacuum. On this interface the equation degenerates, affecting the control and the regularity of the main quantities. For example, it is well-known that the density can develop jumps singularities, therefore preventing any control of the gradient in L2, opposite to the case of linear diffusion. On the other hand, using the fundamental change of variables of the PME, p = uγ, and studying the equation on the pres-sure rather than the equation on the density, turns out to be very useful when searching for better regularity of the gradient. Nevertheless, since the pressure presents "corners" at the free boundary, it is not possible to bound its laplacian in L2 (uniformly on the entire domain).
For these reasons, we could not straightforwardly apply some of the methods previously used in the literature in the case of linear diffusion. For instance, the result in [5] is based on proving H2-a priori bounds, which do not hold in our case. The authors consider elliptic equations in a domain divided into three subdomains, each one contained into the interior of the other. The coefficients of the second-order terms are assumed to be piecewise continuous with jumps along the interior interfaces. Then, the authors study the limit as the thickness of the interior reinforcement tends to zero. In [33], Sanchez-Palencia studies the same problem in the particular case of a lense-shaped region, Iε, which shrinks to a smooth surface in the
limit, facing also the parabolic case. The approach is based on H1-a priori estimates, namely the L2-boundedness of the gradient of the unknown. Considering the variational formulation of the problem, the author is able to pass to the limit upon applying an extension operator. In fact, if the mobility coefficient in Iε converges to zero proportionally with respect to ε, it is only possible to establish uniform bounds outside of Iε. The extension operator allows to "truncate"
the solution and then "extend" it into Iε reflecting its profile from outside. Therefore, making
use of the uniform control outside of the ε-thickness layer, the author is able to pass to the limit in the variational formulation. Let us also mention that, in the literature, one can find different methods and strategies for reaction-diffusion problems with a thin layer. For instance, in [25], the authors develop a multiscale method which combines classical compactness results based on a priori estimates and weak-strong two-scale convergence results in order to be able to pass to the limit in a thin heterogeneous membrane.
The passage at the limit allows to infer the existence of weak solutions for the effective Problem (2). In the case of linear diffusion, the existence of global weak solutions for the effective problem with the Kedem-Katchalsky conditions is provided by [11]. In particular, the authors prove it under weaker hypothesis such as L1 initial data and reaction terms with sub-quadratic growth in an L1-setting.
Outline of the paper. The paper is organised as follows. In Section 2, we introduce the assumptions and notations, including the definition of weak solution of the original problem, System (1). In Section3, a priori estimates that will be useful to pass to the limit are proven.
Section4is devoted to prove the convergence of Problem (1), following the method introduced in [33] for the (non-degenerate) elliptic and parabolic cases. The argument relies on recovering
the L2-boundedness (uniform with respect to ε) of the velocity field, in our case, the pressure gradient. As one may expect, since the permeability of the membrane, µ2,ε, tends to zero proportionally with respect to ε, it is only possible to establish a uniform bound outside of Ω2,ε.
For this reason, following [33], we introduce an extension operator (Subsection 4.1) and apply it to the pressure in order to extend the H1-uniform bounds in the whole space Ω \ ˜Γ1,3, hence
proving compactness results. We remark that the main difference between the strategy in [33] and our adaptation, is given by the fact that due to the non-linearity of the equation, we have to infer strong compactness of the pressure (and consequently of the density) in order to pass to the limit in the variational formulation. For this reason, we also need the L1-boundedness of the time derivative, hence obtaining compactness with a standard Sobolev’s embedding argument. Moreover, since the solution to the limit Problem (2) will present discontinuities at the effective interface, we need to build proper test functions which belong to H1(Ω \ ˜Γ1,3) that are zero on
∂Ω and are discontinuous across ˜Γ1,3, (Subsection 4.2).
Finally, using the compactness obtained thanks to the extension operator, we are able to prove the convergence of a solution to Problem (1) to a couple (˜u, ˜p) which satisfies Problem (2) in a weak sense, therefore inferring the existence of a solution of the effective problem, as stated in the following theorem.
Theorem 1.1 (Convergence to the effective problem). The solution of Problem (1) converges to a solution (˜u, ˜p) of Problem (2) in the following weak sense
− Z T 0 Z Ω ˜ u∂tw + ˜µ1 Z T 0 Z ˜ Ω1 ˜ u∇˜p · ∇w + ˜µ3 Z T 0 Z ˜ Ω3 ˜ u∇˜p · ∇w + ˜µ1,3 Z T 0 Z ˜ Γ1,3 JΠK w|x3=0+− w|x3=0− = Z T 0 Z Ω ˜ uG(˜p)w + Z Ω ˜ u0w0, (6)
for all test functions w(t, x) with a proper regularity and w(T, x) = 0, a.e. in Ω (defined in Theorem4.3), and with
JΠK := γ γ + 1(˜u γ+1) |x3=0+ − γ γ + 1(˜u γ+1) |x3=0−,
and (·)|x3=0± = T (·), with T the trace operator defined in Section2.
Section5 concludes the paper and provides some research perspectives.
2
Assumptions and notations
Here, we detail the problem setting and assumptions. For the sake of simplicity, we consider as domain Ω ⊂ R3 a cylinder with axis x3, see Figure1. Let us notice that it is possible to take
a more general domain ˆΩ defining a proper diffeomorfism F : ˆΩ → Ω. Therefore, the results of this work extend to more general domains as long as the existence of the map F can be proved (this implies that ˆΩ is a connected open subset of Rd and has a smooth boundary). We also want to emphasize the fact that our proofs hold in a 2D domain considering three rectangular subdomains. We introduce uε := u1,ε, in Ω1,ε, u2,ε, in Ω2,ε, u3,ε, in Ω3,ε, pε := p1,ε, in Ω1,ε, p2,ε, in Ω2,ε, p3,ε, in Ω3,ε.
We define the interfaces between the domains Ωi,ε and Ωi+1,ε for i = 1, 2, as
We denote with ni,i+1 the outward normal to Γi,i+1,ε with respect to Ωi,ε, for i = 1, 2. Let us notice that ni,i+1 = −ni+1,i.
We call T the bounded linear trace operator, [4], T : Wk,p(Ω \ ˜Γ1,3) −→ Wk− 1 p,p(∂Ω ∪ ˜Γ 1,3) ⊂ Lp(∂Ω ∪ ˜Γ1,3), for 1 ≤ p ≤ +∞, k > 1 p. Given z ∈ Wk,p(Ω \ ˜Γ1,3), we denote z|∂Ω ∪ ˜ Γ1,3 := T z ∈ W k−1p,p
(∂Ω ∪ ˜Γ1,3) and the following
continuity property holds
kT zk
Wk− 1p ,p(∂Ω ∪ ˜Γ 1,3)
≤ CkzkWk,p(Ω\˜Γ 1,3).
We make the following assumptions on the initial data: there exists a positive constant pH,
such that
0 ≤ p0ε ≤ pH, 0 ≤ u0ε≤ p1/γH =: uH, (A-data1)
∆ (u0i,ε)γ+1 ∈ L1(Ωi,ε), for i = 1, 2, 3. (A-data2)
Moreover, we assume that there exists a function ˜u0∈ L1+(Ω) such that
ku0ε− ˜u0kL1(Ω) −→ 0, as ε → 0. (A-data3)
The growth rate G(·) satisfies
G(0) = GM > 0, G0(·) < 0, G(pH) = 0. (A-G)
The value pH, called homeostatic pressure, represents the lowest level of pressure that prevents
cell multiplication due to contact-inhibition.
We assume that the diffusion coefficients satisfy µi,ε> 0 for i = 1, 3 and lim
ε→0µ1,ε= ˜µ1 > 0, ε→0lim
µ2,ε
ε = ˜µ1,3> 0, ε→0limµ3,ε= ˜µ3> 0. (7)
Notations. For all T > 0, we denote ΩT := (0, T ) × Ω. We use the abbreviated form
uε := uε(t) := uε(t, x). From now on, we use C to indicate a generic positive constant
inde-pendent of ε that may change from line to line. Moreover, we denote sign+(w) = 1{w>0}, sign−(w) = −1{w<0},
and
sign(w) = sign+(w) + sign−(w).
We also define the positive and negative part of w as follows |w|+:= ( w, for w > 0, 0, for w ≤ 0, and |w|−:= ( −w, for w < 0, 0, for w ≥ 0. Now, let us write the variational formulation of Problem (1).
Definition 2.1 (Definition of weak solutions). Given ε > 0, a weak solution to Problem (1) is given by uε, pε∈ L1(0, T ; L∞(Ω)) such that ∇pε∈ L2(ΩT) and
− Z T 0 Z Ω uε∂tψ + 3 X i=1 µi,ε Z T 0 Z Ωi,ε ui,ε∇pi,ε· ∇ψ = Z T 0 Z Ω uεG(pε)ψ + Z Ω u0εψ(0, x), (8) for all test functions ψ ∈ H1(0, T ; H01(Ω)) such that ψ(T, x) = 0 a.e. in Ω.
Remark 2.2. The existence of such solutions can be proved by a straightforward application of standard techniques for nonlinear parabolic equations (see, for instance, [36]).
3
A priori estimates
We show that the main quantities satisfy some uniform a priori estimates which will later allow us to prove strong compactness and pass to the limit.
Lemma 3.1 (A priori estimates). Given the assumptions in Section2, let (uε, pε) be the solution
of Problem (1). The following properties hold uniformly with respect to ε. (i) 0 ≤ uε≤ uH and 0 ≤ pε≤ pH,
(ii) uε, pε ∈ L∞(0, T ; Lp(Ω)), for 1 ≤ p < ∞,
(iii) ∂tuε, ∂tpε ∈ L∞(0, T ; L1(Ω)),
(iv) ∇pε∈ L2(0, T ; L2(Ω \ Ω 2,ε)).
Proof. Let us recall the equation satisfied by uε on Ωi,ε, namely
∂tui,ε− µi,ε∇ · (ui,ε∇uγi,ε) = ui,εG(pi,ε). (9)
(i) 0 ≤ uε ≤ uH, 0 ≤ pε≤ pH. The L∞-bounds of the density and the pressure are a
straight-forward consequence of the comparison principle applied to Equation (9), which can be rewritten as
∂tui,ε−
γ
γ + 1µi,ε∆u
γ+1
i,ε = ui,εG(pi,ε). (10)
Indeed, summing up Equations (10) for i = 1, 2, 3, we obtain
3 X i=1 ∂tui,ε− γ γ + 1 3 X i=1
µi,ε∆uγ+1i,ε = 3
X
i=1
ui,εG(pi,ε). (11)
Then, we also have
3 X i=1 ∂t(uH− ui,ε) = γ γ + 1 3 X i=1
µi,ε∆(uγ+1H − uγ+1i,ε ) + 3 X i=1 (uH − ui,ε)G(pi,ε) − uH 3 X i=1 G(pi,ε).
If we multiply by sign−(uH− ui,ε), thanks to Kato’s inequality, we infer that 3 X i=1 ∂t(uH − ui,ε)−≤ 3 X i=1 γ γ + 1µi,ε∆(u γ+1 H − u γ+1
i,ε )−+ (uH− ui,ε)−G(pi,ε)
− uHG(pi,ε)sign−(uH − ui,ε)
i ≤ 3 X i=1 γ γ + 1µi,ε∆(u γ+1 H − u γ+1
i,ε )−+ (uH− ui,ε)−G(pi,ε)
,
(12)
where we have used the assumption (A-G). We integrate over the domain Ω. Thanks to the boundary conditions in System (1), i.e. the density and flux continuity across the interfaces, and
the homogeneous Dirichlet conditions on ∂Ω, we gain 3 X i=1 Z Ωi,ε
µi,ε∆(uγ+1H − uγ+1i,ε )−
= 2 X i=1 Z Γi,i+1,ε h
µi∇(uγ+1H − uγ+1i,ε )−− µi+1,ε∇(uγ+1H − uγ+1i+1,ε)−
i · ni,i+1 = 2 X i=1 " Z
Γi,i+1,ε∩{uH<ui,ε}
µi∇uγ+1i,ε · ni,i+1−
Z
Γi,i+1,ε∩{uH<ui+1,ε}
µi+1,ε∇uγ+1i+1,ε· ni,i+1
# = 2 X i=1 Z
Γi,i+1,ε∩{uH<ui,ε}
h
µi∇uγ+1i,ε − µi+1,ε∇uγ+1i+1,ε
i
· ni,i+1
= 0.
Hence, from Equation (12), we find d dt 3 X i=1 Z Ωi,ε (uH − ui,ε)−≤ GM 3 X i=1 Z Ωi,ε (uH − ui,ε)−.
Finally, Gronwall’s lemma and hypothesis (A-data1) on u0
i,ε imply 3 X i=1 Z Ωi,ε (uH − ui,ε)−≤ eGMt 3 X i=1 Z Ωi,ε (uH − u0i,ε)−= 0.
We then conclude the boundedness of ui,ε by uH for all i = 1, 2, 3. From the relation pε = uγε,
we conclude the boundedness of pε.
(ii) uε ∈ L∞(0, T ; Lp(Ω)). Integrating Equation (9) in space and taking the sum over all i
we find d dt 3 X i=1 Z Ωi,ε ui,ε− 3 X i=1 Z Ωi,ε
µi,ε∇ · (ui,ε∇pi,ε) = 3 X i=1 Z Ωi,ε ui,εG(pi,ε).
By the boundary and the interface conditions in System (1), upon integrating by parts the second integral turns out to be zero. In fact, we obtain
3
X
i=1
Z
Ωi,ε
µi,ε∇ · (ui,ε∇pi,ε) =
Z Γ1,2,ε µ1,εu1,ε∇p1,ε· n1,2+ Z Γ1,2,ε µ2,εu2,ε∇p2,ε· n2,1 + Z Γ2,3,ε µ2,εu2,ε∇p2,ε· n2,3+ Z Γ2,3,ε µ3,εu3,ε∇p3,ε· n3,2,
which vanishes thanks to the continuity condition on the flux, and the fact that ni,i+1 = −ni+1,i,
for i = 1, 2.
Thanks to the assumptions on G, cf. Equation (A-G), we find d dt 3 X i=1 Z Ωi,ε ui,ε≤ GM 3 X i=1 Z Ωi,ε ui,ε,
and applying Gronwall’s inequality, we obtain
3 X i=1 Z Ωi,ε ui,ε(t) ≤ eGMt 3 X i=1 Z Ωi,ε u0i,ε.
Thanks to the L1 initial bound, we obtain uε∈ L∞(0, T ; L1(Ω)).
Then, since pε = p(γ−1)/γε uε ≤ pH(γ−1)/γ uε, we conclude that pε ∈ L∞(0, T ; L1(Ω)). Let us
notice that since uε and pε are uniformly bounded in L∞, then uε, pε∈ L∞(0, T ; Lp(Ω)), for all
1 ≤ p ≤ +∞.
(iii) ∂tuε, ∂tpε ∈ L∞(0, T ; L1(Ω)). We derive Equation (10) with respect to time to obtain
∂t(∂tui,ε) = µi,εγ∆(pi,ε∂tui,ε) + ∂tui,εG(pi,ε) + ui,εG0(pi,ε)∂tpi,ε.
Upon multiplying by sign(∂tui,ε) and using Kato’s inequality, we have
∂t(|∂tui,ε|) ≤ µi,εγ∆(pi,ε|∂tui,ε|) + |∂tui,ε|G(pi,ε) + ui,εG0(pi,ε)|∂tpi,ε|,
since ui,ε and pi,ε are both nonnegative and ∂tpi,ε= γuγ−1i,ε ∂tui,ε. We integrate over Ωi,ε and we
sum over i = 1, 2, 3, namely d dt 3 X i=1 Z Ωi,ε |∂tui,ε| ≤ γ 3 X i=1 µi,ε Z Ωi,ε ∆(pi,ε|∂tui,ε|) | {z } J +GM Z Ωi,ε |∂tui,ε|, (13)
where we use that G0 ≤ 0.
Now we show that the term J vanishes. Integration by parts yields
J = 2 X i=1 Z Γi,i+1,ε
µi,ε∇(pi,ε|∂tui,ε|) · ni,i+1+ 2
X
i=1
Z
Γi,i+1,ε
µi+1,ε∇(pi+1,ε|∂tui+1,ε|) · ni+1,i.
For the sake of simplicity, we denote n := ni,i+1. Let us recall that, by definition, ni+1,i = −n.
We have J = 2 X i=1 Z Γi,i+1,ε
(µi,ε∇(pi,ε|∂tui,ε|) − µi+1,ε∇(pi+1,ε|∂tui+1,ε|)) · n
= 2 X i=1 Z Γi,i+1,ε
|∂tui,ε|µi,ε∇pi,ε· n − |∂tui+1,ε|µi+1,ε∇pi+1,ε· n
| {z } J1 + 2 X i=1 Z Γi,i+1,ε
µi,εpi,ε∇|∂tui,ε| · n − µi+1,εpi+1,ε∇|∂tui+1,ε| · n
| {z }
J2
.
Let us recall the membrane conditions of Problem (1), namely
µi,εui,ε∇pi,ε· n = µi+1,εui+1,ε∇pi+1,ε· n, (14)
ui,ε= ui+1,ε, (15)
on (0, T ) × Γi,i+1,ε, for i = 1, 2. From Equation (15), it is immediate to infer
∂tui,ε= ∂tui+1,ε, on (0, T ) × Γi,i+1,ε, (16)
since
on Γi,i+1,ε for all h > 0 such that t + h ∈ (0, T ). Combing Equation (15) and Equation (14) we get
µi,ε∇pi,ε· n = µi+1,ε∇pi+1,ε· n on (0, T ) × Γi,i+1,ε. (17)
Moreover, Equation (14) also implies
µi,εpi,ε∇ui,ε· n = µi+1,εpi+1,ε∇ui+1,ε· n on (0, T ) × Γi,i+1,ε, (18)
which, combined with Equation (15) gives also
µi,ε∇ui,ε· n = µi+1,ε∇ui+1,ε· n on (0, T ) × Γi,i+1,ε. (19)
Now we may come back to the computation of the term J . By Equations (16), and (17) we directly infer that J1 vanishes.
We rewrite the term J2 as 2
X
i=1
Z
Γi,i+1,ε
µi,εpi,ε sign(∂tui,ε) ∂t(∇ui,ε· n) − µi+1,εpi+1,ε sign(∂tui+1,ε) ∂t(∇ui+1,ε· n)
= 2 X i=1 Z Γi,i+1,ε
sign(∂tui,ε) ∂t(µi,εpi,ε∇ui,ε· n − µi+1,εpi+1,ε∇ui+1,ε· n)
| {z } J2,1 − 2 X i=1 Z Γi,i+1,ε
|∂tpi,ε|(µi,ε∇ui,ε· n − µi+1,ε∇ui+1,ε· n)
| {z }
J2,2
,
where we used Equation (16), which also implies ∂tpi,ε= ∂tpi+1,εon (0, T ) × Γi,i+1,ε, for i = 1, 2.
The terms J2,1 and J2,2 vanish thanks to Equation (18) and Equation (19), respectively.
Hence, from Equation (13), we finally have
d dt 3 X i=1 Z Ωi,ε |∂tui,ε| ≤ GM 3 X i=1 Z Ωi,ε |∂tui,ε|,
and, using Gronwall’s inequality, we obtain
3 X i=1 Z Ωi,ε |∂tui,ε(t)| ≤ eGMt 3 X i=1 Z Ωi,ε |(∂tui,ε)0|.
Thanks to the assumptions on the initial data, cf. Equation (A-data2), we conclude. (iv) pε∈ L2(0, T ; H1(Ω \ Ω
2,ε)). As known, in the context of a filtration equation, we can
recover the pressure equation upon multiplying the equation on ui,ε, cf. System (1), by p0(ui,ε) =
γuγ−1i,ε . Therefore, we obtain
∂tpi,ε− γµi,εpi,ε∆pi,ε= µi,ε|∇pi,ε|2+ γpi,εG(pi,ε). (20)
Studying the equation on pε rather than the equation on uε turns out to be very useful in order
to prove compactness, since, as it is well-know for the porous medium equation (PME), the gradient of the pressure can be easily bounded in L2, while the density solution of the PME can develop jump singularities on the free boundary, [35].
We integrate Equation (20) on each Ωi,ε, and we sum over all i to obtain 3 X i=1 Z Ωi,ε ∂tpi,ε= 3 X i=1 γµi,ε Z Ωi,ε pi,ε∆pi,ε+ Z Ωi,ε µi,ε|∇pi,ε|2+ γ Z Ωi,ε pi,εG(pi,ε) ! . (21)
Integration by parts yields
3 X i=1 µi,ε Z Ωi,ε pi,ε∆pi,ε= − 3 X i=1 µi,ε Z Ωi,ε |∇pi,ε|2+ 2 X i=1 Z Γi,i+1,ε
µi,εpi,ε∇pi,ε· ni,i+1
+ 2 X i=1 Z Γi,i+1,ε
µi+1,εpi+1,ε∇pi+1,ε· ni+1,i
= − 3 X i=1 µi,ε Z Ωi,ε |∇pi,ε|2,
since we have homogeneous Dirichlet boundary conditions on ∂Ω and the flux continuity condi-tions (17).
Hence, from Equation (21), we have
3 X i=1 Z Ωi,ε ∂tpi,ε= 3 X i=1 µi,ε (1 − γ) Z Ωi,ε |∇pi,ε|2+ γ Z Ωi,ε pi,εG(pi,ε) ! . (22)
We integrate over time and we deduce that
3 X i=1 Z Ωi,ε pi,ε(T ) − Z Ωi,ε p0i,ε+ µi,ε(γ − 1) Z T 0 Z Ωi,ε |∇pi,ε|2 ! = 3 X i=1 γ Z T 0 Z Ωi,ε pi,εG(pi,ε). (23)
Finally, we conclude that
3 X i=1 Z T 0 Z Ωi,ε µi,ε|∇pi,ε|2≤ 3 X i=1 γ γ − 1 Z T 0 Z Ωi,ε pi,εG(pi,ε) + 1 γ − 1 Z Ωi,ε p0i,ε, (24)
Since we have already proved that pi,εis bounded in L∞(ΩT) and by assumption G is continuous,
we finally find that
3 X i=1 µi,ε Z T 0 Z Ωi,ε |∇pi,ε|2 ≤ C, (25)
where C denotes a constant independent of ε. Since both µ1,ε and µ3,εare bounded from below
away from zero, we conclude that the uniform bound holds in Ω \ Ω2,ε.
Remark 3.2. Let us also notice that, differently from [33], where the author studies the linear and uniformly parabolic case, proving weak compactness is not enough. Indeed, due to the presence of the nonlinear term u∇p, it is necessary to infer strong compactness of u. For this reason, the L1-uniform estimate on the time derivative proven in Lemma 3.1 is fundamental.
4
Limit ε → 0
We have now the a priori tools to face the limit ε → 0. We need to construct an extension operator with the aim of controlling uniformly, with respect to ε, the pressure gradient in L2(Ω). Indeed, from (25), we see that one cannot find a uniform bound for k∇p2,εkL2(Ω
2,ε). The
blow-up of Estimate (25) for i = 2, is in fact the main challenge in order to find compactness on Ω. To this end, following [33], we introduce in Subsection 4.1 an extension operator which projects the points of Ω1,ε∪ Ω3,ε inside Ω2,ε. Then, introducing proper test functions such that the variational formulation for ε > 0 in (8) and ε → 0 in (6) are well-defined, we can pass to the limit (Subsection 4.2).
4.1 Extension operator and compactness
Figure 2: Representation of the spatial symmetry used in the definition of the extension operator, cf. Equation (26) and of the two subdomains of Ω2,1,ε and Ω2,3,ε.
As mentioned above, in order to be able to pass to the limit ε → 0, we first need to define the following extension operator
Pε : Lq(0, T ; W1,p(Ω \ Ω2,ε)) → Lq(0, T ; W1,p(Ω \ ˜Γ1,3)), for 1 ≤ p, q ≤ +∞,
as follows for a general function z ∈ Lq(0, T ; W1,p(Ω \ Ω 2,ε)), Pε(z(t, x)) = ( z(t, x), if x ∈ Ω1,ε∪ Ω3,ε, z(t, x0), if x ∈ Ω2,ε, (26)
where x0 is the symmetric of x with respect to Γ1,2,ε (or Γ2,3,ε) if x ∈ Ω2,1,ε (respectively
x ∈ Ω2,3,ε), as illustrated in Figure 2. It can be easily seen that the function g : x → x0 for
x ∈ Ω2,ε and its inverse have uniformly bounded first derivatives. Hence, we infer that Pε is
linear and bounded, i.e.
kPε(z)kLq(0,T ;W1,p(Ω\˜Γ
1,3))≤ C, ∀z ∈ L
q(0, T ; W1,p(Ω \ Ω 2,ε)).
Since the previous bound is uniform in ε, we obtain that Pε remains bounded even in the limit ε → 0. Let us notice that the extension operator is well defined also from L1((0, T ) × (Ω \ Ω2,ε))
Remark 4.1. Thanks to the properties of the extension operator, the estimates stated in Lemma3.1hold true also upon applying Pε(·) on pε, uε, and ∂tpε, namely
0 ≤ Pε(pε) ≤ pH, 0 ≤ Pε(uε) ≤ uH,
∂tPε(pε) ∈ L∞(0, T ; L1(Ω \ ˜Γ1,3)),
∇Pε(pε) ∈ L2(0, T ; L2(Ω \ ˜Γ1,3)).
Lemma 4.2 (Compactness of the extension operator). Let (uε, pε) be the solution of
Prob-lem (1). There exists a couple (˜u, ˜p) with ˜
u ∈ L∞(0, T ; L∞(Ω \ ˜Γ1,3)), p ∈ L˜ 2(0, T ; H1(Ω \ ˜Γ1,3)) ∩ L∞(0, T ; L∞(Ω \ ˜Γ1,3)),
such that, up to a subsequence, it holds
(i) Pε(pε) → ˜p strongly in Lp(0, T ; Lp(Ω \ ˜Γ1,3)), for 1 ≤ p < +∞,
(ii) Pε(uε) → ˜u strongly in Lp(0, T ; Lp(Ω \ ˜Γ1,3)), for 1 ≤ p < +∞,
(iii) ∇Pε(pε) * ∇˜p weakly in L2(0, T ; L2(Ω \ ˜Γ1,3)).
Proof. (i). Since both ∂tPε(pε) and ∇Pε(pε) are bounded in L1(0, T ; L1(Ω \ ˜Γ1,3)) uniformly
with respect to ε, we infer the strong compactness of Pε(pε) in L1(0, T ; L1(Ω \ ˜Γ1,3)). Let us
also notice that since both uε and pε are uniformly bounded in L∞(0, T ; L∞(Ω \ ˜Γ1,3)) then the
strong convergence holds in any Lp(0, T ; Lp(Ω \ ˜Γ1,3)) with 1 ≤ p < ∞.
(ii). From (i), we can extract a subsequence of Pε(pε) which converges almost everywhere.
Then, remembering that uε = p1/γε , with γ > 1 fixed, we have convergence of Pε(uε) almost
everywhere. Thanks to the uniform L∞-bound of Pε(uε), Lebesgue’s theorem implies the
state-ment. Let us point out that, in particular, the L∞-uniform bound is also valid in the limit. (iii). The uniform boundedness of ∇Pε(pε) in L2(0, T ; L2(Ω \ ˜Γ1,3)) immediately implies
weak convergence up to a subsequence.
4.2 Test function space and passage to the limit ε → 0
Since in the limit we expect a discontinuity of the density on ˜Γ1,3, we need to define a suitable
space of test functions. Therefore we construct the space E? as follows. Let us consider a function ζ ∈ D(Ω) (i.e. Cc∞(Ω)). For any ε > 0 small enough, we build the function vε= Pε(ζ),
using the extension operator previously defined. The space of all finite (and therefore well-defined) linear combinations of these functions vε is called E? ⊂ H1(Ω \ ˜Γ1,3). We stress that
the functions of E? are discontinuous on ˜Γ1,3.
In the weak formulation of the limit problem (6), we will make use of piece-wise C∞-test functions (discontinuous on ˜Γ1,3) of the type w(t, x) = ϕ(t)v(x), where ϕ ∈ C1([0, T )) with
ϕ(T ) = 0 and v ∈ E∗. Therefore, w belongs to C1([0, T ); E∗). On the other hand, in the variational formulation (8), i.e. for ε > 0, H1(0, T ; H01(Ω)) test functions are required. Thus, in order to study the limit ε → 0, we need to introduce a proper sequence of test functions depending on ε that converges to w. To this end, we define the operator Lε : C1([0, T ); E∗) → H1(0, T ; H01(Ω)) such that
Lε(w) → w, uniformly as ε → 0, ∀w ∈ C1([0, T ); E?).
In this way, Lε(w) belongs to H1(0, T ; H01(Ω)), therefore, it can be used as test function in the formulation (8).
Following Sanchez-Palancia, [33], for all t ∈ [0, T ] and x = (x1, x2, x3) ∈ Ω, we define Lε(w(t, x)) = w(t, x), if x /∈ Ω2ε, 1 2 h w t, x1, x2, ε 2 + w t, x1, x2, − ε 2 i + h w t, x1, x2, ε 2 − wt, x1, x2, − ε 2 ix3 ε, otherwise.
It can be easily verified that Lε is linear with respect to x3 in Ω2,ε and is continuous on ∂Ω2,ε. Let us notice that it holds
∂Lε(w) ∂x3 ≤ C ε. (27)
Furthermore, thanks to the mean value theorem, the partial derivatives of Lε with respect to x1
and x2 are bounded by a constant (independent of ε), ∂Lε(w) ∂x1 ≤ C, ∂Lε(w) ∂x2 ≤ C, and since the measure of Ω2,ε is proportional to ε, we have
Z T 0 Z Ω2,ε ∂Lε(w) ∂x1 2 + ∂Lε(w) ∂x2 2 ≤ Cε. (28)
Given w ∈ C1([0, T ); E?), we take Lε(w) as a test function in the variational formulation of the
problem, i.e. Equation (8), and we have
− Z T 0 Z Ω uε∂tLε(w) + 3 X i=1 µi,ε Z T 0 Z Ωi,ε ui,ε∇pi,ε· ∇Lε(w) = Z T 0 Z Ω uεG(pε)Lε(w) + Z Ω u0εLε(w0). (29)
Thanks to the a priori estimates already proven, cf. Lemma3.1, Remark4.1and the convergence result on the extension operator, cf. Lemma4.2, we are now able to pass to the limit ε → 0 and recover the effective interface problem.
Theorem 4.3. For all test functions of the form w(t, x) := ϕ(t)v(x) with ϕ ∈ C1([0, T )) and v ∈ E∗, the limit couple (˜u, ˜p) of Lemma 4.2satisfies the following equation
− Z T 0 Z Ω ˜ u∂tw + ˜µ1 Z T 0 Z ˜ Ω1 ˜ u∇˜p · ∇w + ˜µ3 Z T 0 Z ˜ Ω3 ˜ u∇˜p · ∇w + ˜µ1,3 Z T 0 Z ˜ Γ1,3 JΠK w|x3=0+− w|x3=0− = Z T 0 Z Ω ˜ uG(˜p)w + Z Ω ˜ u0w0, where JΠK := γ γ + 1(˜u γ+1) |x3=0+ − γ γ + 1(˜u γ+1) |x3=0−, and (·)|x
3=0± = T (·), with T the trace operator defined in Section2.
Step 1. Time derivative integral. We split the first integral into two parts − Z T 0 Z Ω uε∂tLε(w) = − Z T 0 Z Ω1,ε∪Ω3,ε uε∂tLε(w) | {z } I1 − Z T 0 Z Ω2,ε uε∂tLε(w) | {z } I2 .
Since outside of Ω2,εthe extension operator coincides with the identity, and Lε(w) = w, we have I1 = − Z T 0 Z Ω1,ε∪Ω3,ε Pε(uε)∂tw = − Z T 0 Z Ω Pε(uε)∂tw + Z T 0 Z Ω2,ε Pε(uε)∂tw.
Thanks to Remark4.1, we know that the last integral converges to zero, since both Pε(uε) and
∂tw are bounded in L2 and the measure of Ω2,ε tends to zero as ε → 0. Then, by Lemma 4.2,
we have − Z T 0 Z Ω Pε(uε)∂tw −→ − Z T 0 Z Ω ˜ u ∂tw, as ε → 0,
where we used the weak convergence of Pε(uε) to ˜u in L2(0, T ; L2(Ω \ ˜Γ1,3)). The term I2
vanishes in the limit, since both uε and ∂tLε(w) are bounded in L2 uniformly with respect to ε.
Hence, we finally have − Z T 0 Z Ω uε∂tLε(w) −→ − Z T 0 Z Ω ˜ u ∂tw, as ε → 0. (30)
Step 2. Reaction integral. We use the same argument for the reaction term, namely Z T 0 Z Ω uεG(pε)Lε(w) = Z T 0 Z Ω1,ε∪Ω3,ε uεG(pε)Lε(w) | {z } K1 + Z T 0 Z Ω2,ε uεG(pε)Lε(w) | {z } K2 .
Using again the convergence result on the extension operator, cf. Lemma4.2, we obtain
K1= Z T 0 Z Ω1,ε∪Ω3,ε Pε(uε)G(Pε(pε))w −→ Z T 0 Z Ω ˜ u G(˜p)w, as ε → 0,
since both Pε(uε) and G(Pε(pε)) converge strongly in L2(0, T ; L2(Ω \ ˜Γ1,3)). Arguing as before,
it is immediate to see that K2 vanishes in the limit. Hence
Z T 0 Z Ω uεG(pε)Lε(w) −→ Z T 0 Z Ω ˜ uG(˜p)w, as ε → 0. (31)
Step 3. Initial data integral. From (A-data3), it is easy to see that Z Ω u0εLε(w0) −→ Z Ω ˜ u0w0, as ε → 0. (32)
Step 4. Divergence integral. Now it remains to treat the divergence term in Equation (29), from which we recover the effective interface conditions at the limit.
Since the extension operator Pε is in fact the identity operator on Ω \ Ω2,ε, we can write
3 X i=1 µi,ε Z T 0 Z Ωi,ε ui,ε∇pi,ε· ∇Lε(w) = X i=1,3 µi,ε Z T 0 Z Ωi,ε Pε(ui,ε)∇Pε(pi,ε) · ∇w | {z } H1 + µ2,ε Z T 0 Z Ω2,ε u2,ε∇p2,ε· ∇Lε(w) | {z } H2 . (33)
We treat the two terms separately. Since we want to use the weak convergence of ∇Pε(pε) in
L2(0, T ; L2(Ω \ ˜Γ
1,3)) (together with the strong convergence of Pε(uε) in L2(0, T ; L2(Ω \ ˜Γ1,3)))
we need to write the term H1 as an integral over Ω. To this end, let µε:= µε(x) be a function
defined as follows µε(x) := µ1,ε for x ∈ Ω1,ε, 0 for x ∈ Ω2,ε, µ3,ε for x ∈ Ω3,ε.
Then, we can write
H1 = Z T 0 Z Ω µεPε(uε)∇Pε(pε) · ∇w.
Let us notice that as ε goes to 0, µεconverges to ˜µ1in ˜Ω1and ˜µ3in ˜Ω3. Therefore, by Lemma4.2,
we infer H1−→ ˜µ1 Z ˜ Ω1 ˜ u ∇˜p · ∇w + ˜µ3 Z ˜ Ω3 ˜ u ∇˜p · ∇w, as ε → 0. (34)
Now we treat the term H2, which can be written as H2 =µ2,ε Z T 0 Z Ω2,ε u2,ε∇p2,ε· ∇Lε(w) =µ2,ε Z T 0 Z Ω2,ε u2,ε ∂p2,ε ∂x1 ∂Lε(w) ∂x1 + u2,ε ∂p2,ε ∂x2 ∂Lε(w) ∂x2 + µ2,ε Z T 0 Z Ω2,ε u2,ε ∂p2,ε ∂x3 ∂Lε(w) ∂x3 . By the Cauchy-Schwarz inequality, the a priori estimate (25), and Equation (28), we have
µ2,ε Z T 0 Z Ω2,ε u2,ε ∂p2,ε ∂x1 ∂Lε(w) ∂x1 + u2,ε ∂p2,ε ∂x2 ∂Lε(w) ∂x2 ≤ µ1/22,εku2,εkL∞((0,T )×Ω 2,ε) µ1/22,ε ∂p2,ε ∂x1 L2((0,T )×Ω 2,ε) ∂Lε(w) ∂x1 L2((0,T )×Ω 2,ε) ! + µ1/22,εku2,εkL∞((0,T )×Ω 2,ε) µ1/22,ε∂p2,ε ∂x2 L2((0,T )×Ω 2,ε) ∂Lε(w) ∂x2 L2((0,T )×Ω 2,ε) ! ≤ C µ1/22,ε ε1/2→ 0.
On the other hand, by Fubini’s theorem, the following equality holds µ2,ε Z T 0 Z Ω2,ε u2,ε ∂p2,ε ∂x3 ∂Lε(w) ∂x3 = µ2,ε γ γ + 1 Z T 0 Z Ω2,ε ∂uγ+12,ε ∂x3 ∂Lε(w) ∂x3 = µ2,ε γ γ + 1 Z T 0 Z ε/2 −ε/2 Z ˜ Γ1,3 ∂uγ+12,ε ∂x3 ∂Lε(w) ∂x3 dσ dx3 = µ2,ε γ γ + 1 Z T 0 Z ε/2 −ε/2 Z ˜ Γ1,3 ∂uγ+12,ε ∂x3 w|x3=ε2 − w|x3=−ε2 ε dσ dx3 = µ2,ε ε γ γ + 1 Z T 0 Z ˜ Γ1,3 w|x3=ε2 − w|x3=−ε2 Z ε/2 −ε/2 ∂uγ+12,ε ∂x3 dx3dσ = µ2,ε ε γ γ + 1 Z T 0 Z ˜ Γ1,3 (uγ+12,ε )|x3=ε2 − (u γ+1 2,ε )|x3=−ε2 ·w|x3=ε2 − w|x3=−ε2 . Therefore, lim ε→0H2 = limε→0 µ2,ε ε γ γ + 1 Z T 0 Z ˜ Γ1,3 (uγ+12,ε )|x3=ε2 − (u γ+1 2,ε )|x3=−ε2 ·w|x3=ε2 − w|x3=−ε2 . (35) In order to conclude the proof, we state the following lemma, which is provenbelow.
Lemma 4.4. The following limit holds uniformly in ˜Γ1,3
w|x3=ε2 − w|x3=−ε2 −→ w|x3=0+ − w|x3=0−, as ε → 0. (36) Moreover, γ γ + 1 (uγ+12,ε )|x3=ε2 − (u γ+1 2,ε )|x3=−ε2 −→ γ γ + 1 (˜u γ+1) |x3=0+− (˜u γ+1) |x3=0−, (37) strongly in L2(0, T ; L2(˜Γ1,3)), as ε → 0.
We may finally find the limit of the term H2, using Assumption (7), and applying Lemma4.4
to Equation (35) µ2,ε ε γ γ + 1 Z T 0 Z ˜ Γ1,3 (uγ+12,ε )|x3=ε2 − (u γ+1 2,ε )|x3=−2ε ·w|x3=ε2 − w|x3=−ε2 −→ ˜µ1,3 γ γ + 1 Z T 0 Z ˜ Γ1,3 ˜ uγ+1|x 3=0+ − ˜u γ+1 |x3=0− · w|x3=0+− w|x 3=0−,
as ε → 0. Combining the above convergence to Equation (33) and Equation (34), we find the limit of the divergence term as ε goes to 0,
3 X i=1 µi,ε Z T 0 Z Ωi,ε ui,ε∇pi,ε· ∇Lε(w) −→ ˜µ1 Z T 0 Z ˜ Ω1 ˜ u∇˜p · ∇w + ˜µ3 Z T 0 Z ˜ Ω3 ˜ u∇˜p · ∇w + ˜µ1,3 γ γ + 1 Z T 0 Z ˜ Γ1,3 (˜uγ+1)|x3=0+− (˜u γ+1) |x3=0− · w|x3=0+− w|x3=0−,
which, together with Equations (29), (30), (31), and (32), concludes the proof.
We now turn to the proof of Lemma4.4
Proof of Lemma 4.4. Since by definition w(t, x) = ϕ(t)v(x), with ϕ ∈ C1([0, T )) and v ∈ E∗, the uniform convergence in Equation (36) comes from the piece-wise differentiability of w.
A little bit trickier is the second convergence, i.e. Equation (37). We recall that on {x3= ±ε/2}, uγ+12,ε coincides with Pε(uγ+1ε ), since across the interfaces uε is continuous and
Pε(ui,ε) = ui,ε, for i = 1, 3. Moreover, from (i) and (iv) in Remark 4.1, we infer that
γ γ + 1∇ Pε(u γ+1 ε ) = Pε(uε)∇Pε(pε) ∈ L2(0, T ; L2(Ω \ ˜Γ1,3)), hence P(uγ+1ε ) L2(0,T ;H1(Ω\˜Γ 1,3)) ≤ C.
Again from Remark4.1, we know that ∂t Pε(uγ+1ε ) L∞(0,T ;L1(Ω\˜Γ 1,3))≤ C.
Since we have the following embeddings
H1(Ω \ ˜Γ1,3) ⊂⊂ Hβ(Ω \ ˜Γ1,3) ⊂ L1(Ω \ ˜Γ1,3),
for every 12 < β < 1, upon applying Aubin-Lions lemma, [1,23], we obtain Pε(uγ+1ε ) −→ ˜uγ+1, as ε → 0,
strongly in L2(0, T ; Hβ(Ω \ ˜Γ1,3)).
Thanks to the continuity of the trace operator T : Hβ(Ω \ ˜Γ1,3) → L2(∂Ω ∪ ˜Γ1,3), for 1
2 < β < 1, we finally recover that
Pε(u γ+1 ε )|x3=0±− ˜u γ+1 |x3=0± L2(0,T ;L2(˜Γ 1,3)) ≤ C Pε(uγ+1ε ) − ˜uγ+1 L2(0,T ;Hβ(Ω\˜Γ 1,3))→ 0, (38) as ε → 0. We recall that the trace vanishes on the external boundary, ∂Ω, therefore we only consider the L2(0, T ; L2(˜Γ1,3))-norm. Recalling that L is the length of Ω, trivially, we find the
following estimate Pε(uγ+1ε )|x3=±ε/2− Pε(u γ+1 ε )|x3=0± 2 L2(0,T ;L2(˜Γ 1,3)) = Z T 0 Z ˜ Γ1,3 Z ±ε/2 0 ∂Pε(uγ+1ε ) ∂x3 !2 = Z T 0 Z ˜ Γ1,3 Z L ∂Pε(uγ+1ε ) ∂x3 1[0,±ε/2] (x3) !2 ≤ Z T 0 Z ˜ Γ1,3 Z L ∂Pε(uγ+1ε ) ∂x3 !2 Z L 1[0,±ε/2] (x3) 2 ≤ ε 2k∇Pε(u γ+1 ε )kL2(0,T ;L2(Ω\˜Γ 1,3)) ≤ ε C,
Remark 4.5. Although not relevant from a biological point of view, let us point out that, in the case of dimension greater than 3, the analysis goes through without major changes. It is clear that the a priori estimates are not affected by the shape or the dimension of the domain (although some uniform constants C may depend on the dimension, this does not change the result in Lemma 3.1). The following methods, and in particular the definition of the extension operator and the functional space of test functions, clearly depends on the dimension, but the strategy is analogous for a d-dimensional cylinder with axis {x1 = · · · = xd−1= 0}.
5
Conclusions and perspectives
We proved the convergence of a continuous model of cell invasion through a membrane when its thickness is converging to zero, hence giving a rigorous derivation of the effective transmission conditions already conjectured in Chaplain et al., [10]. Our strategy relies on the methods developed in [33], although we had to handle the difficulties coming from the nonlinearity and degeneracy of the system. We did not consider the case of non-constant mobilities, i.e. µi,ε := µi,ε(t, x), which could bring additional challenges to the derivation of the a priori estimates
and the compactness results. In particular, a very interesting direction both from the biological and mathematical point of view, could be coupling the system to an equation describing the evolution of the MMP concentration. In fact, as observed in [10], the permeability coefficient can depend on the local concentration of MMPs, since it indicates the level of "aggressiveness" at which the tumour is able to destroy the membrane and invade the tissue.
In a recent work [15], a formal derivation of the multi-species effective problem has been proposed. However, its rigorous proof remains an interesting and challenging open question. Indeed, introducing multiple species of cells, hence dealing with a cross-(nonlinear)-diffusion system, adds several challenges to the problem. As it is well-known, proving the existence of solutions to cross-diffusion systems with different mobilities is one of the most challenging and still open questions in the field. Nevertheless, even when dealing with the same constant mobility coefficients, the nature of the multi-species system (at least for dimension greater than one) usually requires strong compactness on the pressure gradient. We refer the reader to [17,29] for existence results of the two-species model without membrane conditions.
Another direction of further investigation of the effective transmission problem (2) could be studying the so-called incompressible limit, namely the limit of the system as γ → ∞. The study of this limit has a long history of applications to tumour growth models, and has attracted a lot of interest since it links density-based models to a geometrical (or free boundary) representation, cf. [19,27].
Moreover, including the heterogeneity of the membrane in the model could not only be useful in order to improve the biological relevance of the model, but could bring interesting mathematical challenges, forcing to develop new methods or adapt already existent ones, [25], from the parabolic to the degenerate case.
Acknowledgements
G.C. and A.P. have received funding from the European Research Council (ERC) under the Eu-ropean Union’s Horizon 2020 research and innovation programme (grant agreement No 740623). The work of G.C. was also partially supported by GNAMPA-INdAM.
N.D. has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie (grant agreement No 754362).
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